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Topic 13 Practice Paper: Circle theorems — question paper

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QQ/IGCSE-MATHS/T13

IGCSE · Cambridge IGCSE Mathematics (Extended (Core + Extended))

Topic 13 Practice Paper

Circle theorems

Duration: 40 minutes
Total: 30 marks
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READ THESE INSTRUCTIONS FIRST

  • Answer all questions.
  • Section A contains ten multiple-choice questions. Choose one answer, A, B, C or D, for each.
  • Section B contains structured questions. Write your answers in the spaces provided.
  • The number of marks is given in brackets [ ] at the end of each question or part question.
  • The total mark for this paper is 30.

Original practice material written against the Cambridge IGCSE Mathematics syllabus. It is not an official Cambridge examination paper. Topic 13: the angle in a semicircle, tangent–radius, angle at the centre, same-segment angles, cyclic quadrilaterals, the alternate segment theorem, equal tangents, and chord bisection.

For Examiner’s Use
Section A/ 10
Section B/ 20
Total/ 30

Section A: Multiple-choice questions[10 marks]

Choose one answer for each question. Every correct answer scores one mark.

  1. 1

    With AB as a diameter and C anywhere on the rim, angle ACB measures

    1. A45°
    2. B60°
    3. C90°
    4. D180°
    ( )
  2. 2

    A tangent touches a circle at T. The angle between the tangent and the radius OT is

    1. A90°
    2. B45°
    3. C60°
    4. D180°
    ( )
  3. 3

    A centre angle of 100° means the matching rim angle on that arc is (Extended)

    1. A100°
    2. B50°
    3. C200°
    4. D25°
    ( )
  4. 4

    A rim angle of 35° on arc XY doubles at the centre to (Extended)

    1. A17.5°
    2. B140°
    3. C35°
    4. D70°
    ( )
  5. 5

    In a cyclic quadrilateral, the corner facing an 89° angle holds (Extended)

    1. A91°
    2. B89°
    3. C271°
    4. D101°
    ( )
  6. 6

    Rim angles from the same chord, in the same segment: one is 27°, so the other is (Extended)

    1. A153°
    2. B54°
    3. C27°
    4. D63°
    ( )
  7. 7

    One tangent from P has length 11 cm. The other tangent from P is (Extended)

    1. A5.5 cm
    2. B11 cm
    3. C22 cm
    4. D121 cm
    ( )
  8. 8

    Triangle OAB is built from two radii with angle AOB = 84°; its base angles measure (Extended)

    1. A84°
    2. B96°
    3. C42°
    4. D48°
    ( )
  9. 9

    A tangent and chord meet at 71°; across the chord, the alternate segment holds (Extended)

    1. A109°
    2. B19°
    3. C71°
    4. D142°
    ( )
  10. 10

    Tangents from P touch at X and Y, and angle XOY = 150°. Angle XPY is (Extended)

    1. A150°
    2. B30°
    3. C75°
    4. D210°
    ( )

Section B: Structured questions[20 marks]

Answer all questions in the spaces provided. Show all your working.

  1. 11

    C perches on the major arc AB while D sits on the minor arc; the centre angle AOB reads 130°. (Extended)

    1. (a)

      Deduce angle ACB.

      [2]
    2. (b)

      Deduce angle ADB.

      [2]
    3. (c)

      Verify that angles ACB and ADB fit a circle theorem.

      [1]
  2. 12

    Take AB as a diameter and place C and D on the rim; angle CAB reads 34°.

    1. (a)

      Write down angle ACB, with a reason.

      [2]
    2. (b)

      Find angle ABC.

      [2]
    3. (c)

      Write down angle ADB.

      [1]
  3. 13

    Two tangent lines from P kiss the circle at X and Y (centre O), with the 70° gap XPY between them. (Extended)

    1. (a)

      Give the size of angles OXP and OYP, and say why.

      [1]
    2. (b)

      Work out angle XOY.

      [2]
    3. (c)

      Explain why PX = PY.

      [1]
    4. (d)

      Given OP = 13 cm and PX = 12 cm, find the radius of the circle.

      [2]
  4. 14

    Where the tangent at A meets chord AC, the angle is 57°. B sits in the alternate segment, D in the opposite one. (Extended)

    1. (a)

      Find angle ABC.

      [2]
    2. (b)

      Find angle ADC.

      [2]

— End of paper —